Add material and apply Diffuse color via Python
Actually, I figured it out shortly after posting the question.

Instead of using:

bpy.ops.material.new()


I needed to use:

bpy.data.materials.new(name="MaterialName")


So the final solution was:

activeObject = bpy.context.active_object #Set active object to variable
mat = bpy.data.materials.new(name="MaterialName") #set new material to variable
activeObject.data.materials.append(mat) #add the material to the object
bpy.context.object.active_material.diffuse_color = (1, 0, 0) #change color